# Interesting Harmonic Oscillator Solution

On page 89 of Griffith's QM book, an exact solution to the time-dependent SE equation for the harmonic oscillator is mentioned:

$$\Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right]$$

He says that this solution was found by Schrodinger in 1927. I am wondering if anyone can tell me where this solution comes from or at least point me to a paper where it is discussed. I haven't been able to find much on it. Thank you for any help.

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I have heard his solutions came from simple "series solution" to differential equation. Not sure about references though. – jerk_dadt Mar 8 '14 at 6:32
I was thinking that it had come from the time evolution of some initial wave function. Maybe a Gaussian displaced from the origin? – TylerHG Mar 8 '14 at 7:39
It does have a Gaussian feel to it. However it has the same constants outside and inside the exponential as the stationary solution. His stationary solution came from series solution method (I believe Hermite polynomials), so I was thinking series solution to a partial differential equation. Its odd that there it is a exponential of complex exponential. Exponential inception. – jerk_dadt Mar 8 '14 at 8:11
@TylerHG: It is indeed a Gaussian, but with an oscillatory displacement: $\cos(\omega t)$ (look at $|\psi|^2$). – Kyle Kanos Mar 8 '14 at 17:22